Exercise set 2.2

Throughout, K denotes a field and V a vector space over K of finite dimension d ≥ 1. We fix an integer m ≥ 1 and denote by G the group GL_K(m). We denote by AK(m) the ring of polynomial functions on G and by SK(m) and SK(m,n) the appropriate Schur algebras (as defined in the lecture).

(1)

(Difficulty level 2) What is the dimension of the K-algebra SK(m,n)?

(2)

(Difficulty level 1) Show that G → SK(m) given by gδ_g is an injection. (Here and elsewhere, the “Dirac delta” δ_g denotes the linear functional on AK(m) given by ff(g).)

(3)

(Difficulty level 2) Show that δ_g ⋆ δ_h = δ_gh for g and h in G.

(4)

(Difficulty level 3) Recall the following three definitions of the multiplication of two elements α and β in SK(m). Convince yourself that they are the same:

(a)

(α ⋆ β)(f) = f(xy)dα(x)dβ(y)

(b)

(α ⋆ β)(f) = β(yα(ρ_yf)), where ρ_yf(x) := f(xy).

(c)

(α ⋆ β)(f) = ∑ _iα(f_i¹)β(f_i²), where Δ(f) = ∑ _if_i¹ ⊗ f_i² is the image of f under the “coproduct” Δ, which is the K-algebra map from AK(m) → AK(m) ⊗ AK(m) given by X_ij∑ _kX_ik ⊗ X_kj

(5)

(Difficulty level 2) Show that under the multiplication defined as in the previous item, the Schur algebra SK(m) becomes an associative unital K-algebra. What is its multiplicative identity?

(6)

(Difficulty level 2) Recall that the inclusion of SK(m,n) in SK(m) is induced by the natural surjection of AK(m) onto its n-th degree component (which maps any polynomial to its homogeneous component of degree n). Show that SK(m,n) is a two-sided ideal of SK(m).

(7)

(Difficulty level 2) By Sym(V ) we mean the polynomial algebra generated in the variables z₁, …, z_d, where e₁, …, e_d form a basis of V . Note that Sym(V ) is a polynomial ring with the usual grading (where the variables z₁, …, z_d are all considered to be of degree 1). The graded piece of degree n is denoted by Symⁿ(V ). There is a natural action of GL(V ) on Sym(V ) by algebra automorphisms that preserves degrees: for g in GL(V ), we let gz_j = a₁(g)z₁ + … + a_d(g)z_d, where ge_j = a₁(g)e₁ + … + a_d(g)e_d (in other words, the action on the variables is the same as that on e₁, …, e_d). The Symⁿ(V ) provide examples of polynomial representations of GL(V ).